Reference : Alternate modules are subsymplectic
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Physical, chemical, mathematical & earth Sciences : Mathematics
Alternate modules are subsymplectic
Guerin, Clément mailto [University of Luxembourg > Faculty of Science, Technology and Communication (FSTC) > Mathematics Research Unit >]
[en] alternate modules ; Lagrangians ; Finite Abelian groups
[en] In this paper, an alternate module $(A,\phi)$ is a finite abelian group $A$ with a $\mathbb{Z}$-bilinear application $\phi:A\times A\rightarrow \mathbb{Q}/\mathbb{Z}$ which is alternate (i.e. zero on the diagonal). We shall prove that any alternate module is subsymplectic, i.e. if $(A,\phi)$ has a Lagrangian of cardinal $n$ then there exists an abelian group $B$ of order $n$ such that $(A,\phi)$ is a submodule of the standard symplectic module $B\times B^*$.
This paper won't be submitted per-se. It proves a technical result with elementary methods. It is used in my work on centralizers in PSL(n,C). It will eventually become an appendix of this paper.

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